https://file.aiquickdraw.com/mathbot/file/1725826119759qsk60uil.pdf

Let's work on **Question 2** from the file you uploaded:

### Given matrix:
\[
M = \begin{pmatrix} k + 1 & k + 1 \\ k + 1 & k + 2 \end{pmatrix}
\]

### Step 1: Formula for the inverse of a 2x2 matrix
The inverse of a 2x2 matrix \( M = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \) is given by:
\[
M^{-1} = \frac{1}{ad - bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}
\]
This requires finding the determinant of matrix \( M \), which is \( ad - bc \).

### Step 2: Determinant of the matrix
The determinant of matrix \( M \) is:
\[
\text{det}(M) = (k+1)(k+2) - (k+1)(k+1)
\]
Simplifying this expression:
\[
\text{det}(M) = (k+1)[(k+2) - (k+1)] = (k+1)(1) = k+1
\]

### Step 3: Inverse of the matrix
Since the determinant is \( k+1 \), the inverse of the matrix \( M \) is:
\[
M^{-1} = \frac{1}{k+1} \begin{pmatrix} k+2 & -(k+1) \\ -(k+1) & k+1 \end{pmatrix}
\]
Simplifying this gives:
\[
M^{-1} = \frac{1}{k+1} \begin{pmatrix} k+2 & -(k+1) \\ -(k+1) & k+1 \end{pmatrix}
\]

This is the inverse of the matrix \( M \) in terms of \( k \).

Would you like more details on this solution or have any questions?

### Relative Questions:
1. How do you verify the inverse of a matrix is correct?
2. What happens when \( k = -1 \) for this matrix?
3. Can we find the inverse if the determinant is zero?
4. What are some applications of matrix inversion in real-world problems?
5. How would you calculate the inverse of a 3x3 matrix?

**Tip**: Always check the determinant before computing the inverse, as a zero determinant indicates the matrix is non-invertible.

Find, in terms of k, the inverse of the following 2x2 matrix: M = [[k+1, k+1], [k+1, k+2]].

Learn how to find the inverse of a 2x2 matrix M = [[k+1, k+1], [k+1, k+2]] in terms of k. This solution includes key concepts in matrix algebra, such as determinants and matrix inversion. Suitable for high school students.

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